## D03. Define the following terms: monomer, polymer, weight-average molecular weight and number-average molecular weight.

Repetitive units make up a polymer. These repetitive units were originally the monomer molecules.

When polymer chains form their lengths and thus their weights differ. It is important to be able to characterize the polymer structure. Determining the weight-average molecular weight or the number-average molecular weight is a part of any polymer characterization.

Degree of polymerization

The smallest repeatable unit in a polymer molecule is the mer unit (monomer). The size of a molecule is determined by dividing the molecular weight by the mer weight. This number is called the degree of polymerization, DP:

DP = (molecular weight)/(mer weight)

The dimensions for our calculation are = (amu/molecule)/(amu/mer) = mers/molecule

As an example, a polyvinyl chloride molecule may have a molecular weight of 35,000 amu (or 35,000 g/mol). Since the unit weight for C2H3Cl is 2 * (12 amu) + 3 * (1 amu) + 35.5 amu =  62.5 amu, the degree of polymerization is 35,000 amu/62.5 amu = 560 mers/molecule. In commercial plastics the degree of polymerization normally falls in the range of 75 to 750 mers per molecule.

The molecule described above has a molecular weight of 35,000. Such a value differs by a major order of magnitude from that of other molecules. However, as large as the polymer molecule appears to be by weight, it is still smaller than the resolving power of an optical microscope, and only under certain circumstances can it be resolved even by an electron microscope. Consequently, molecular weight determinations are usually made indirectly by such physical means as measurement of viscosity, osmotic pressure, or light scattering, all of which are affected by the number, size, or shape of molecules in a suspension or in a solution.

Average molecular weight

When a material like polyethylene or polyvinyl chloride is formed from small molecules, not all the resulting large molecules are identical in size. As might be expected, some grow larger than others do. As a result, plastics contain a range of molecular sizes, somewhat analogous to the mixture of propane, hexane, octane, and other, paraffin hydrocarbons in crude oils. Hence it is necessary to calculate the average degree of polymerization if a single index is desired.

1. Weight-average molecular weight

One procedure for determining average molecular weight utilizes the weight fraction of the polymer that is in each of several size fractions (Table 1). The "weight-average" molecular weight, Mw, is calculated as follows:

Mw = {Sum [(Wi)(MW)i] }/{Sum Wi }

where Wi is the weight fraction of each size fraction and (MW)i is the mean molecular weight of the size fraction. The "weight-average" molecular weight is particularly significant in the analysis of properties such as viscosity, where the weight of the molecules is important.

(See example 1)

2. Number-average molecular weight

Properties such as strength are more sensitive to the numbers of molecules in each weight fraction than to the actual weight. Hence, a "number-average" molecular weight, Mn, has some significance:

Mn ={Sum [(Wi)(MW)i]}/{Sum Xi}

The value Xi is the number of molecules in each size fraction, and is calculated as shown in Example 2.

(See example 2)

To conclude

Weight- average molecular weight is larger or equal to number-average molecular weight.

Weight- average molecular weight and molecular weight distributions are determined from ultracentrifuge sedimentation, diffusion and light scattering. Number-average molecular weight and molecular weight distributions are determined from osmotic pressure and intrinsic viscosity determinations.

Optical properties are best reflected in the weight-average molecular weight, while strength properties are best reflected in number-average molecular weight.

E-mail: ksoderholm@dental.ufl.edu